transactions of the american mathematical society Volume 314, Number 2, August 1989
THE NONLINEAR GEOMETRY OF LINEAR PROGRAMMING. II LEGENDRE TRANSFORM COORDINATES AND CENTRAL TRAJECTORIES
D. A. BAYER AND J. C. LAGARIAS
Abstract. Karmarkar's projective scaling algorithm for solving linear program- ming problems associates to each objective function a vector field defined in the interior of the polytope of feasible solutions of the problem. This paper studies the set of trajectories obtained by integrating this vector field, called P-trajectories, as well as a related set of trajectories, called A-trajectories. The /1-trajectories arise from another linear programming algorithm, the affine scal- ing algorithm. The affine and projective scaling vector fields are each defined for linear programs of a special form, called standard form and canonical form, respectively. These trajectories are studied using a nonlinear change of variables called Legendre transform coordinates, which is a projection of the gradient of a loga- rithmic barrier function. The Legendre transform coordinate mapping is given by rational functions, and its inverse mapping is algebraic. It depends only on the constraints of the linear program, and is a one-to-one mapping for canoni- cal form linear programs. When the polytope of feasible solutions is bounded, there is a unique point mapping to zero, called the center. The .4-trajectories of standard form linear programs are linearized by the Legendre transform coordinate mapping. When the polytope of feasible so- lutions is bounded, they are the complete set of geodesies of a Riemannian geometry isometric to Euclidean geometry. Each /1-trajectory is part of a real algebraic curve. Each P-trajectory for a canonical form linear program lies in a plane in Legendre transform coordinates. The P-trajectory through 0 in Legendre transform coordinates, called the central P-trajectory, is part of a straight line, and is contained in the ^-trajectory through 0 , called the central A-trajectory. Each /"-trajectory is part of a real algebraic curve. The central ^-trajectory is the locus of centers of a family of linear programs obtained by adding an extra equality constraint of the form (c, x) = p . It is also the set of minima of a parametrized family of logarithmic barrier functions. Power-series expansions are derived for the central .4-trajectory, which is also the central P-trajectory. These power-series have a simple recursive form and are useful in developing "higher-order" analogues of Karmarkar's algorithm. ^-trajectories are defined for a general linear program. Using this definition, it is shown that the limit point Xoo of a central ^-trajectory on the boundary of the feasible solution polytope P is the center of the unique face of P con- taining Xoo in its relative interior.
Received by the editors October 8, 1986 and, in revised form, June 9, 1987 and March 25, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 90C05; Secondary 52A40, 34A34. The first author was partially supported by ONR contract N00014-87-K0214.
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The central trajectory of a combined primal-dual linear program has a simple set of polynomial relations determining it as an algebraic curve. These relations are a relaxed form of the complementary slackness conditions. This central trajectory algebraically projects onto the central trajectories of both the primal and dual linear programs, and this gives an algebraic correspondence between points on the positive parts of the central trajectories of the primal and dual linear programs. Two Lagrangian dynamical systems with simple Lagrangians are shown to have /1-trajectories as q-trajectories. The Hamiltonian dynamical systems as- sociated to these Lagrangian systems are completely integrable.
1. Introduction In 1984 Narendra Karmarkar [K] introduced a new linear programming algo- rithm which he proved ran in polynomial time in the worst case. This algorithm, which we call the projective scaling algorithm, takes a series of piecewise linear steps in the relative interior of the polytope of feasible solutions of the linear programming problem. The step direction is computed using a vector field de- fined on the relative interior of the polytope of feasible solutions. This vector field, which we call the projective scaling vector field, depends on the linear pro- gram's constraints and on the objective function. The projective scaling vector field is defined for linear programs of a special form which we call canonical form, and Karmarkar uses projective transformations to compute this vector field direction. Our viewpoint is that a fundamental object underlying the projective scaling algorithm is the set of trajectories obtained by integrating this vector field, which we call projective scaling trajectories, or P-trajectories, and that the polynomial- time nature of Karmarkar's algorithm arises from special geometric properties of these trajectories. The projective scaling algorithm provides one method of approximately fol- lowing these trajectories. Several authors [Re, V, Go, KMY] have recently developed other linear programming algorithms that follow these trajectories in a different manner. This series of papers studies the P-trajectories and a related family of curves which we call affine scaling trajectories or A-trajectories for short. The affine scaling trajectories are associated to another vector field, the affine scaling vector field, which arises in connection with another interior-point linear programming method, the affine scaling algorithm, which was originally proposed by 1.1. Dikin [Dl, D2] in 1967, and rediscovered by many people including [B, VMF]. In part I we defined the affine and projective scaling vector fields and showed that /'-trajectories of a canonical form linear program: ' minimize (c, x), ^x = 0, (e, x) = n , x>0,
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having e = (1,1, ... , l)T as a feasible solution were the radial projections of ^-trajectories of the homogeneous standard form linear program:
{minimize (c, x), Ax = 0, x>0,
whose polytope of feasible solutions is a cone. Consequently P-trajectories can be studied by studying the ^-trajectories of a related linear programming problem. This paper studies these trajectories using a nonlinear change of coordinates which we call Legendre transform coordinates. Legendre transform coordinates are given by a projection of the gradient of a logarithmic barrier function deter- mined by the linear program's constraints. We show that Legendre transform coordinates linearize the ^-trajectories. As a consequence when the polytope P of feasible solutions is bounded the set of ^-trajectories for all objective functions forms a complete set of geodesies for a Riemannian geometry on the relative interior of P which is isometric to Euclidean geometry. P-trajectories are partially linearized by Legendre transform coordinates in the sense that each P-trajectory lies in a plane in Legendre transform coordinates. (This is explained further in part III, where it is shown that P-trajectories are linearized by a related mapping, projective Legendre transform coordinates.) The Legen- dre transform coordinate mapping is a rational mapping, and consequently the ^-trajectories are parts of real algebraic curves. P-trajectories are also proved to be parts of real algebraic curves. We use Legendre transform coordinates to derive power-series expansions for ^-trajectories. Associated to any set of constraints H having a bounded polytope PH of feasible solutions is a special point xH which we call the center of PH . It is the point mapped to 0 in Legendre transform coordinates, and coincides with the "analytical center" of Sonnevend [Sol, So2]. For each objective function we call the ^-trajectory (resp. P-trajectory) through the center the central A-trajectory (resp. central P-trajectory). We show the central P-trajectory coincides with the central yi-trajectory. We give several other characterizations of the central ^-trajectory, one of which is that it is the set of solutions to a set of fixed point problems of a parametrized family of logarithmic barrier functions, the logarithmic barrier function trajectory. We describe interpretations due to the second author of ^-trajectories as q-trajectories of Lagrangian dynamical systems whose associated Hamiltonian dynamical systems are completely integrable. We are indebted to Mike Todd for many comments improving the exposition of this paper. The results of §§1-9 were presented at MSRI in January 1986.
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2. Summary of results The affine scaling vector fields and projective scaling vector fields are defined for linear programs of special forms which we call standard form and canonical form, respectively. A standard form linear program is one of the following form: {minimize (c, x), ,4x = b, x>0. We call the constraints of such a linear program standard form constraints. We say such a linear program is in strict standard form if it has a feasible solution x = (xx, ... ,xn) with all x, > 0, and we call the corresponding constraints strict standard form constraints. Strict standard form constraints have the nice property that the relative interior Rel-Int(P) of the polytope P of feasible solutions has Rel-Int(P) = PnInt(R"), so consists exactly of vectors x in P with x > 0. A canonical form linear program is one of the following form: minimize (c, x), Ax = 0, (2.2) (e,x) = n, x>0, such that e = ( 1,1, ... , 1) is a feasible solution, i.e., Ae = 0. Constraints of this form are called canonical form constraints. A canonical form linear program is a special kind of strict standard form linear program. The affine and projective scaling vector fields are defined using affine and projective transformations to rescale the linear programs, in a manner described in part I, §4. The affine scaling vector field \A(x;c) for the objective function (c, x) at an interior feasible point x of the strict standard form linear program (2.1) is given by vA(x;c) = -Xn{AX)±(Xc) where ' x,M X = diag(x)
Xn is a diagonal matrix and x,AX)± denotes orthogonal projection onto the sub- space (AX) . In particular n(AX)± = I - XAT(AX2AT)~XAX. The projective scaling vector field \p(x;c) for the objective function (c,x) atan interior feasible point x of a canonical form linear program (2.2) is given by (2.4) yp(x;c) = -Xn{AX)±(Xc) + (l/n)(Xe,n{AX)±(Xc))Xe.
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The affine scaling trajectories and projective scaling trajectories are obtained by integrating these vector fields. The affine scaling trajectory or A-trajectory TA(x;c,A,b) of the standard form linear program (2.1) is the point-set deter- mined by the solution to the affine scaling differential equation: (dx(t)/dt = vA(x(t);c), \x(0) = x, extended to the maximal range of the parameter t for which a solution is defined. The projective scaling trajectory, or P-trajectory Tp(x;c,A) of the canonical form problem (2.2) is the point-set determined by the projective scal- ing differential equation: (dx(t)/dt = \p(x(t);c), \x(0) = x, extended to the maximal range of t for which a solution is defined. §3 introduces a nonlinear change of variables, Legendre transform coordi- nates, to study ^-trajectories and P-trajectories. The Legendre transform co- ordinate mapping where L T denotes the transpose of L. §3 also gives an explicit formula for the Legendre transform coordinates For strict standard form problems the mapping 0H(x) is one-to-one, and its range is the relative interior of the cone CH = -nA±(R"+), where R" is the positive orthant in R" . License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 532 D. A. BAYER AND J. C. LAGARIAS §4 determines the effect on tangent vectors of the Legendre transform map- ping for a strict standard form linear program. The tangent spaces in the domain and range can both be identified with Ax and we show for v e A± that at the point x one has d(f>H(\) = nA±(X~2y), and d d(f>H(vA(x;c)) = -nA±(c), so that the affine scaling vector field is constant. Hence the images d This vector field has a constant component in the direction c* = tt[4]X(c) and a component pointing radially. As a consequence each P-trajectory 4>(Tp(x;c,A)) in Legendre transform coordinates lies in a plane in the Legendre transform coordinate space. A set of linear program constraints H having a bounded polytope PH of feasible solutions has a unique point xH whose Legendre transform coordinates are 4>H(xH) = 0. We call this point the center of PH . This notion coincides with the "analytical center" introduced by Sonnevend [Sol, So2]. The central A-trajectory for a given linear program having a bounded poly- tope of feasible solutions is that ^-trajectory passing through the center of License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE NONLINEAR GEOMETRY OF LINEAR PROGRAMMING. II 533 PH . More generally, for a standard form problem (2.1) we define the central A-trajectory TA(c; A, b) for the objective function (c,x) by TA(c;A,b) = {x: (Ax = b, \x>0, together with the extra equality constraint (c,x) = p, for the values p~ < p < p+ , where p.f and p~ are the maximum and min- imum of the objective function (c,x) on the polytope of feasible solutions, respectively. Third, if the feasible solution polytope is bounded then that part of the central ,4-trajectory of a strict standard form linear program on one side on the center is the logarithmic barrier function trajectory, described as the set x(p) of solutions to the following parametrized family of nonlinear minimiza- tion problems: minimize (c,x) -p j ^logx; 1 , Ax = b, lx>0, for 0 < p < oo . §7 derives two power-series expansions for ,4-trajectories. The first power- series expansion applies to an arbitrary ^-trajectory of a standard form problem and takes as power-series parameter a Legendre transform coordinate parameter t, defined by 0H(X(Z)) = /(^)7r[^]x(c) , for a certain scalar function f(z). These power-series expansions have a sim- ple form, suitable for computation, which yield "higher-order" analogues of Karmarkar's algorithm, cf. [AKRV, KLSW]. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 534 D. A. BAYER AND J. C LAGARIAS §8 defines ^-trajectories and central /1-trajectories for a general linear pro- gram on R" given in the inequality form n „ j minimize (c,x), ( ] \(aj,x)>bj; l where v is a fixed vector and the parameter p varies. §8 gives three different definitions of ^-trajectories and proves they are all equivalent. §9 studies central trajectories for linear programs having a full-dimensional polytope of feasible solutions. It shows that if H is a set of inequality constraints and H' is another set of constraints obtained by adding extra constraints of the form ± (c, x) > b, for various different values of b, then the central A- trajectory TA(c,w!) attached to H' is contained in the central ,4-trajectory TA(c, H) attached to H . In the case that PH is a bounded polytope, the center of H' is generally in a different place on the central trajectory than the center of H . Reneger [Re] bases a linear programming algorithm on the idea of approximately following a series of such centers along the central trajectory to an optimal point; the series of centers is obtained by adding extra constraints of the form - (c, x) > p where p is a parameter whose value is changed at each step, and the centers are followed using Newton's method. §9 also shows that the limit point x^ of a central /1-trajectory 7^(c,H) on the boundary dPH of the polytope PH of feasible solutions is the center of the unique face of dPH in which x^ is a relative interior point. § 10 relates the central trajectories of a pair of dual linear programming prob- lems. Consider the following pair (P) and (D) of dual linear programs: ( minimize (c, x), ATx>b, and minimize - (b, y), {Ay = c, y>0. The combined primal-dual linear program (PD) is: ' minimize (c, x) - (b, y), ATx>b, (PD) Ay = c, U>0. The central trajectory of (PD) orthogonally projects onto the positive half of the central trajectory of (P) in the x-variables, and onto the positive half of the central trajectory of (D) in the y-variables. This gives rise to a one-to- one algebraic correspondence between points on the positive half of the central License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE NONLINEAR GEOMETRY OF LINEAR PROGRAMMING. II 535 trajectory of (P) and points on the positive half of the central trajectory of (D) (see Theorem 10.1). This correspondence was previously observed by Osbourne [Os] in the context of logarithmic barrier function trajectories. § 11 gives polynomial ideals of relations which specify algebraic curves that contain ^-trajectories. The central ^-trajectory of the combined primal-dual linear program (PD) satisfies a particularly simple set of relations. It is the set of polynomial relations in the polynomial ring C[x,y,u,p] generated by the relations A Tx - u = b, ^y = c, yjUj=p, l License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 536 D. A. BAYER AND J. C. LAGARIAS r(H) = n . Any set of constraints H for which PH is bounded is necessarily of full rank. We first define the Legendre transform coordinate mapping 4>H(x) in the full- dimensional case when Int(PH) is nonempty. Associate to H the logarithmic barrier function m (3.2) /H(x) = -5>g«a,,x>-¿>;), 7 = 1 which is defined for x e Int(PH). The Legendre transform mapping To define the Legendre transform coordinate mapping in the case that PH is nonempty and of dimension less than n , we need some more definitions. A con- straint (a , x) > b, in H is nonsingular if there exists a feasible point xQ with (a, x0) > b, ; otherwise it is singular. (That is, the singular constraints function as equality constraints.) Let Wn denote the set of nonsingular constraints in H; by renumbering the constraints, we may suppose they are (3.4) (a],x)>bj, l (3-5) ^H(x) = ^H(0Hn(x)) = 7rDH(V/Hn(x)) Here nD denotes orthogonal projection onto the feasible direction subspace DH. The Legendre transform coordinate mapping is most naturally viewed as a mapping onto a suitable quotient space of the dual space (R")*. This "coordi- nate-free" definition is given in Appendix A. In it the gradient V/H(x) is re- placed by the differential dfH(x), which is an element of (R")*, the set of linear functional / : Rn -* R. The gradient V/H is obtained from the differential dfH using the (not natural) isomorphism (R")* —»R" given by the transpose map: / e (Rn)* corresponds to y e Rn where /y(x) = (y,x). This mapping also identifies quotient spaces of (Rn)* with subspaces of R" , and quotient map- pings on (R")* with projection operators on R" . We use the coordinatized definition for License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE NONLINEARGEOMETRY OF LINEAR PROGRAMMING.II 537 The basic properties of the Legendre transform mapping in the full-dimen- sional case are as follows. Theorem 3.1. Let H be a set of constraints (Aj ,x)>bj, l has domain Int(PH) andränge Rel-Int(CH), where CH is the cone (3.7) CH = R+[-a,,-a2,-,-aJ. If H is of full rank then CH is full dimensional and (3.8) Rel-Int(CH) = \ x: x = - ¿pjk} with all p; > 0 Formula (3.6) shows that x 6 Int(PH) implies that 4>H(x)e Rel-Int(CH). To show that the mapping is onto, let c e Rel-Int(CH) and write c = -^7,7=1P-j^j ' w^tn au A4,> 0 • Consider the function m (3.9) *(x) = - (3.10) g(x) = ¿ (-Uc,x) - log((a,,x) - bj)) 7=1 V ' and prove that each term in the sum is bounded below on Int(PH). Since m m - License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 538 D. A. BAYER AND J. C LAGARIAS with all p > 0, on letting B = J2ßjb. we have --(c,x)>-y2p.b, = -B, m m ¿-" J J m so that -(c,x) is bounded below. Then since (3.10) gives (3.11) -{c,x)>iij({mj,x)-bj) + B, we have -¿(c,x) -log«a,,x) - bj) > -l(c,x) -log (~{c,x) - jß\ which is bounded below. If W = {x: (c,x) = 0 and (ay,x) = 0 for 1 < ; < m*}, then x0 e PH implies that the flat xQ + W C PH, and g(x) is constant on this flat. Let IV± denote the orthogonal complement of W, and set PH =PHnH/J". Then PH = PH@W and g(x) attains all values in its range on Rel-Int(PH). Finally one checks that g(x) —►oo in PH as x approaches dPH because some -log((a ,x) - b,) —►oo, and g(x) —►oo as ||x|| —►oo in PH because -(c,x) —►co. (The last assertion holds because in PH as ||x|| —►oo either -(c,x) —>oo or else some (a ,x) - b , -* oo, in which case -(c,x) -» oo by (3.11).) This implies that g(x) attains a global minimum in Rel-Int(PH), which is also its global minimum in Int(PH). The claim is proved. If H is full rank, then CH is full dimensional. Also the Jacobian V0H(x) is the Hessian V2/H(x), which is m . (3.12) V^H(x) = V-?a,a[. Each term a a is a positive semidefinite symmetric matrix. The full-rank hypothesis guarantees that V Since R" = [a,, ... ,am] there exists some (a- ,y) > 0 so this gives y V License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE NONLINEAR GEOMETRYOF LINEAR PROGRAMMING. II 539 hyperplanes {x: (a , x) = b,} . In this paper we are concerned with the mapping x = where the minus sign is taken in the square root to give the branch of Int(PH) -^ Int(PJ(H)) (3.14) 4>J Uj(H) lt Rel-Int(CH) *- Rel-Int(CJ(H)) Proof. The relation PJ(H)= J(PH) is immediate. Next we claim that /j(h)(J(x)) = /h(x)> for all x e Int(PH). Indeed if y = J(x) then m /,,H)(JW) = - El0g«ay ' J"'(J(X))) - bj) = M*) ■ 7=1 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 540 D. A. BAYER AND J. C LAGARIAS Now differentials transform contravariantly under invertible affine transforma- tions, i.e., for any function f(x), if f(y) = f(J~X(y)) then (3.15) (df)xW= f((df)J{x))(J(y)), for v a tangent vector at x, where J* : (R")* —►(Rn)* is the linear transforma- tion adjoint to J, defined by J*(/)(x) = /(J(x)-J(0)) = /(Lx), where / e (R")* is a linear functional /:R"-»R. Under the identification of (Rfl)* with R" by /y(x) = (y,x) one has J*(y) = Lry and (df)x = (V/)x. Thus (3.14) follows as a special case of (3.15). D We call the mapping License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE NONLINEAR GEOMETRY OF LINEAR PROGRAMMING. II 541 The Legendre transform coordinate mapping (3.19) Rel-Int(Cïï(i) <- Rel-Int(CHii) ni>» u •Rel-Int(CH) The commutativity of the top square in (3.19) follows from the general trans- formation property of gradients under a one-to-one affine change of variable (which generalizes (3.15)). The bottom triangle in (3.19) commutes because (3.20) Lry = 0 forallyeZ)^. This holds since for all x e R one has Lx e DH so that (LTy ,x) = (y ,Lx) = 0. Now by Theorem 3.1 the map «p^ is one-to-one and onto its range Rel-Int(Crr ), hence the commutativity of (3.19) guarantees that the mapping License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 542 D. A. BAYER AND J. C LAGARIAS Theorem 3.4. Let H be a set of constraints in R" (*j,x)>bj, l on R\ Then DJ{H)= J(DH), PJ(H)= J(PH), and the following diagram com- mutes: Rel-Int(PH) Rel-Int(PJ(H))(»)' (3.21) Rel-Int(CH) Rel-Int(CJ(H)). We remark that although the choice of Jr-l is not necessarily unique, the Legendre transform mapping Proof. We can find a full-dimensional linear program H on R , where d = d(H), and two injective affine mappings J, : R —►R" and J2: R -* R such that J,(H) = H, J2(H) = J(H), and such that the top triangle in the following diagram commutes: Int(Pïï), Rel-Int(Pïï) ►Rel-Int(PJ(H)) t>H 0H Rel-Int(Cïï) Rel-Int(CH) Rel-Int(CJ(H)) Here J,(x) = L,x + m,. Then the whole diagram commutes, for the two sides commute by (3.19), and the relation L L2 = Lx follows from Jo Jj = J2. This proves (3.21). Finally since J and License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE NONLINEAR GEOMETRY OF LINEAR PROGRAMMING. II 543 Given a set of inequality constraints H of full rank the map (3.22) J(xH) = x•J(H) holds for any affine transformation that is one-to-one on the domain PH . In the full-dimensional case the center is defined by V/H (xH) = 0, which is equivalent to the condition that xH maximizes the function m 7=1 on Int(PH), so xH coincides with the "analytical center" of Sonnevend [Sol, So2]. Now we find Legendre transform coordinates for the constraints of a (strict) standard form linear program. Recall that standard form constraints are given by (3.23, {£-/■ and that such a set of constraints is in strict standard form if it has a feasible solution x = (xx, ... ,xn) with all xt > 0. For a linear program in strict standard form the relative interior Rel-Int(PH) of the polytope PH of feasible solutions is nonempty and is PH n Int(R" ), which is (Ax = b, \x>0. A set of standard form constraints H is always of full rank. For a set of strict standard form constraints H the nonsingular constraints Hn are xx> 0, 1 < i < n, and the associated logarithmic barrier function is n ¡=i Hence 1/X. where X = diag(x,, ... ,xn) is a diagonal matrix. Theorem 3.3 yields the following result, stated as a theorem for ease of ref- erence. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 544 D. A. BAYER AND J. C. LAGARIAS Theorem 3.5. Let H denote a set of strict standard form linear programming constraints, iAx = b, \x>0. The Legendre transform mapping (3.24) The mapping (3.25) provided that the matrix A has full row rank so that (AAT)~~Xexists. This formula shows that the entries of is algebraic, and that (dcf>H)x:T(Rel-Int(PH))x^ T(a\ denote the differential of (3.26) (d (3.27) (d License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE NONLINEAR GEOMETRY OF LINEAR PROGRAMMING. II 545 Proof. For v in A and x in Rel-Int(PH) the curve c(t) = x + iv lies in Rel-Int(PH) for -e < t < e with e small enough, and has v as tangent vector at x. Then if X = diag(x) and V = diag(v) then for |r| small enough we have = -nA±(X~xe - tX~2Ve + 0(t2)) = cf>H(x)+ tnA±(X-2y) + 0(t2), from which (3.27) follows. Here we use the identity (X + tV)~X =(i + tx~xv~x)~xx~x which is valid for small enough \t\ since the diagonal matrices X and V com- mute and X is invertible. We define the linear operators L,: R" —>■R" by L,(w) = nA±(X~~2yv)and L2(w) = XnAX±(Xyy). It is easy to check that both these operators have range spaces contained in A . The formula (3.28) for (dcp^ ) is equivalent to the algebraic identity (3.29) L2(Lx(y))=y, all v e AL . That is, the (generally noninvertible) linear operators Lx and L2 when re- stricted to the domain A are invertible and are mutual inverses of each other. To verify (3.29) suppose that v e A and set y = Lx(y) = nAX(X~2y). Using nA± = I - A(AAT)~XA we have (3.30) L2 o L, (v) = L2(v) = Xn,AX)±(Axy (XnA± (X~2y)) Xn,AX^(X'■(ax)±\"- Xy)-Xn.AX,±(XAT(AAT)v) _ "-"-(Axy 'AX \). For the first term on the right of (3.30) we find on expanding n,AX)± that (3.31) Xn(AX)±(X~Xy) = XX~Xy - X2A(AX2ATfXAy = y, using the fact that Ay = 0 since v is in A± . For the second term on the right in (3.30), on expanding n,AX)± we obtain X(XAT(AAT)~XAX~2y) - X(XAT[(AX2ATfXAX(XAT)](AAT)~XAX~2y) = 0, using the fact that the product in brackets is the identity. Combining the last three equations proves (3.29) holds. D License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 546 D. A. BAYERAND J. C LAGARIAS 4. Affine scaling vector field in Legendre transform coordinates We take as given a linear program in standard form: (minimize (c, x), ^x = b, x>0 such that there exists a feasible solution x = (xx, ... ,x ) with all xt > 0. In part I, Lemma 4.1, it is shown that the affine scaling vector field \A(x;c) for a strict standard form linear program is (4.2) yA(x;c) = -Xn{AX)±(Xc), where X = diag(x). The affine scaling vector yA(x ; c) lies in A , as may be verified using (4.2). In part I, Lemma 4.1 it is also shown that the vector field v^(x;c) depends only on the component nA±(c) of c in the ^-direction, i.e., (4.3) yA(x;c) = yA(x;nA±(c)). We have the following result. Theorem 4.1. Given a set H of standard form constraints (Ax = b, \x>0, having a feasible point x with all xj. > 0, so that Rel-Int(PH) = PH n Int(R"). Let v^(x;c) = -Xn{AX,±(Xc) denote the affine scaling vector field associated to the objective function (c,x). If (j>H: Rel-Int(PH) —> A is the Legendre transform map (4.4) (d yA(x;nA±(c)) = -Xn{AX)±(XnA±(c)) = (d where y = (dc/>H)x(yA(x;c)) = (dH)x(yA(x;nA±(c))) = (d proving the theorem. G This result immediately shows that ^-trajectories are parts of straight lines in Legendre transform coordinates. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE NONLINEAR GEOMETRYOF LINEAR PROGRAMMING. II 547 Corollary 4.1a. Let a set H of strict standard form constraints be given, and let TA(x;c) be the affine scaling trajectory associated to the objective function (c,x) that goes through x e Rel-Int(PH). Let dH(xx,x2) = d( License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 548 D. A. BAYERAND J. C. LAGARIAS Corollary 4.1c. Every A-trajectory of a standard form linear programming prob- lem is part of a real algebraic curve in the linear program's coordinate system. Proof. Any standard form linear programming problem H may be brought to strict standard form by dropping all variables x, which are identically zero on PH . Consequently it suffices to prove the corollary for strict standard form linear programs only. For strict standard form problems the result follows di- rectly from Corollary 4.1a, since each ^-trajectory is the inverse image under a rational mapping of a line segment, a half-line, or a straight line. D All P-trajectories are parts of a real algebraic curve, because each P-trajec- tory is algebraically related to an ^4-trajectory of a standard form problem. Corollary 4.1d. Every P-trajectory of a canonical form linear programming prob- lem is part of a real algebraic curve in the linear program's coordinate system. Proof. Part I, Theorem 7.1 showed that any P-trajectory Tp(x;c,A) of a canonical form problem is the radial projection of the ,4-trajectory TA(x;c,A,0) of the associated strict standard form linear program obtained by dropping the constraint (e, x) = n , i.e., Tp(x;c,A) = {ny/(e,y):yeTA(x;c,A,0)}. Since this is an algebraic relation and this /i-trajectory is part of a real algebraic curve by Corollary 4.1c, so is Tp(x;c,A). D 5. Protective scaling vector field in Legendre transform coordinates Consider a linear program in canonical form: ' minimize (c, x), ,. ,. ,4x = 0, (5A) \I (e,x)i \ = n, x>0, where e is a feasible solution. Part I, Lemma 4.3 showed that the projective scal- ing vector field yp(x;c,A) for a canonical form linear program with objective function (c,x) is (5.2) yp(x;c,A) = -Xn{AX)±(Xc) + (l/n)(Xe,n(AX)±(Xc))Xe, where X = diag(x). We normally abbreviate yp(x;c,A) to v/)(x;c), omitting the explicit dependence on the constraints. The projective scaling vector field vp(x;c) for x in Rel-Int(PH) lies in [AT] , as may be verified from (5.2). Part I, Lemma 4.3 showed v/,(x;c) depends only on the component nA±(c) of c in the A -direction, i.e., (5.3) yp(x;c)=yp(x;nA±(c)). This vector field does depend on the component of c in the e-direction, even though this component of the objective function remains constant on the poly- tope PH due to the constraint (c, x) = n. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE NONLINEARGEOMETRY OF LINEAR PROGRAMMING.II 549 Now we compute the projective scaling vector field in Legendre transform coordinates. Theorem 5.1. Let H be a set of canonical form linear programming constraints, and let vp(x;c) be the projective scaling vector field associated to the objective function (c,x) at xe Rel-Int(PH), given by (5.4) vp(x;c) = -Xn[AX)±(Xc) + (l/n)(Xe,n{AX)±(Xc))Xe. If (5.5) H(x)= -n[&]±(X-Xe), then (5.6) (dcpH)x(yp(x;c)) = -n^]±(c) - (l/n)(Xe,n{AX)±(Xc))cf>(x). Proof. Apply Theorem 3.2 to the formula for \p(x;c) above to obtain (5.7) (d4>H)x(yp(x;c)) = - n^(X~ln(AX)X(Xc)) e' + (l/n)(Xe, n± (Xc))n^± (X~Xe). To simplify the first term on the right of (5.9), we use (5.8) X~Xn,AX)±(Xc)-(Axy = X~X(I - XAT(AX2AT)~XAX)Xc c- A w, where (5.9) yy= (AX2AT)'XAX2c. T T Now nA±(A w) = 0, either by noting that A w lies in the row space of A , or by directly calculating nA±(ATyv) = (I - AT(AAT)~XA)AT(AX2ATyXAX2c = 0. Thus (5.8) yields -nljr]±(X~Xn{AX)±(Xc)) = -7t^]±(c) + n{eT)±(nA±(ATyv)) = -nlj.]±(c), using the fact that 7trj,]± = n(tT)±nAi. since Ae = 0. Substituting this equality and the identity (5.5) into (5.7) completes the proof. D We remark that the vector w = (AX A )~ AX c has a natural interpretation as a set of dual variables (see Todd-Burrell [TB]). Theorem 5.1 shows that the projective scaling vector field in Legendre trans- form coordinates is a vector sum of the constant vector field -c* where c* = Ä[j.]j.(c) > together with a vector field that points radially toward or away from the origin 0 at each point in the Legendre transform coordinate space. This has the following consequence. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 550 D. A. BAYER AND J. C. LAGARIAS Corollary 5.1 a. For a canonical form linear program every P-trajectory with objective function (c, x) viewed in the Legendre transform coordinate space lies in a plane containing the line L = {c t : - oo < t < 00}, where c =n,A,± (c). C 1e7~ 1 Proof. Consider the P-trajectory passing through a point y = The projective scaling vector field \p(x;c) depends on the component of c in the e-direction. An objective function (c,x) is normalized if (c,x) > 0 for all feasible solutions and (c,x) = 0 for some feasible solution. Normal- ized objective functions play a special role in Karmarkar's projective scaling algorithm, and Karmarkar's convergence proof only applies to normalized ob- jective functions. The criterion that an objective function (c, x) be normalized uniquely determines its component in the e-direction, as follows (proved in part I, Lemma 4.4). Given any objective function c of a canonical form linear program there is a unique normalized objective function c^ such that c^ lies in A-1, and n^]±(c) = ^^(c^). In fact if c* = ^^(c) and xopt is an optimal solution then c,v = c*-(l/rt)(c\xopt)e. It can be proved that an objective function is normalized if and only if in Legendre transform coordinates all P-trajectories are asymptotically parallel to the central P-trajectory as they approach an optimal solution, see part III. 6. Central trajectories The Legendre transform coordinate map (6.1) 0h(xh) = O> which we call the center of H . By Theorem 3.5 a canonical form linear program always has a bounded polytope PH and the point e is its center. We call the A- trajectory and P-trajectory through the center e of a canonical form problem with objective function (c,x) the central A-trajectory with objective function (c,x), denoted TA(c,A), and the central P-trajectory with objective function (c,x), denoted by Tp(c,A). The trajectory TA(c,A) is TA(e;c,[$],[$]) in the notation of part I. We first show that central ^4-trajectories contain (and usually coincide with) central P-trajectories. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE NONLINEAR GEOMETRY OF LINEAR PROGRAMMING. II 551 Theorem 6.1. The central P-trajectory Tp(c,A) of a canonical form linear pro- gram is contained in the central A-trajectory TA(c,A) for the same linear pro- gram. They coincide if and only if the projective scaling vector field v/,(x;c) does not vanish for all x on the central A-trajectory TA(c,A). Proof. Let c* = n.A „(c). By Corollary 4.1a the central ^-trajectory is simply the line -ct in Legendre transform coordinates, i.e., (6.2) TA(e;c,A) = {x: n[jL]±(X e) =-tc*, - oo < t < oo}. Let x( = Xte denote the unique point on 7^(e;c,.4) with 0H(X() = -n[A,]±-(X7 e) = ~tc* ■ e' By Theorem 5.1 the projective scaling vector field at xr in Legendre transform coordinates is (d License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 552 D. A. BAYER AND J. C LAGARIAS This definition reduces to (6.2) for a canonical form problem, and also makes sense for standard form problems having an unbounded domain PH . The cen- tral ^-trajectory TA(c,A,b) so defined is a curve if c* = nA±(c) e Rel-Int(CH) with c* t¿ 0 ; it is the center {xH} if c* = 0 e Rel-Int(CH) ; it is the empty set if c* i Rel-Int(CH). The central trajectory T^c.^b) is an affine invariant in the sense that if ^d-i (x) = D~ x is an affine transformation with all dt > 0 then (6.6) 4V,(7>,^,b)) = TA(Dc,AD,b). This follows from Theorem 3.2. One characterization of the central ^4-trajectory TA(c,A,b) for a strict stan- dard form linear program is that it is the locus of centers of a parameterized family of standard form linear programs having an extra sliding equality con- straint of the form (c, x) = p added. Theorem 6.2. Let H denote the set of strict standard form linear program con- straints: (Ax = b \x>0 and let H denote these constraints together with the extra constraint (c,x) = p. Set p+ = max((c,x): x e PH) and p_ = min((c,x): x e PH), and suppose p_ < p+ . Let x(p) denote the center of PH if one exists. Then the central trajectory T^c^.b) is the set of centers x(p) of PH for p_ < p < p+, i.e., (6.7) T(c,A,b) = {x(p):p_ Set c = 7tA±(c) and observe that where nèT and nA± commute since Ac = 0. Hence one has (6.9) 4>Hii(x)= -n{èT)±(nA±(X~le)) = n{èT)±( License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE NONLINEAR GEOMETRY OF LINEAR PROGRAMMING. II 553 Using (6.8) one obtains <^(x) = 7f(ê7>L(-&) = 0. Hence H has a center x(p) and x = x(p). This proves that T(c,A,b) ç {x : p_ < p < p+} . For the reverse inclusion if (j>H(x) = 0 then (6.9) yields n{iT)±( (6.10) minimize (c, x) - p ( J^1°S*,-- \) over x e Rel-Int(PH) attains its minimum at a unique point x(p) for 0 < p < oo. The Legendre transform coordinates of x(p) are (6.11) License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 554 £>. A. BAYERAND J. C LAGARIAS so that (6.14) By hypothesis 7tA±(c) / 0, and there is a unique point x(p) in Rel-Int(PH) satisfying (6.14), which is on the positive part of the central trajectory; further- more there is then a unique X such that (6.12) holds, so that x(p) is a critical point. It is the unique global minimum of g (x), and taking 0 < p < oo one obtains all points on the positive side of the center. D Theorem 6.3 holds also for standard form linear programs having an un- bounded polytope PH of feasible solutions, provided that the objective function (c,x) is not constant on PH and that -nA±(c) is in the range Rel-Int(CH) of the Legendre transform coordinate mapping, so that (6.14) is solvable and the minimization problem (6.10) then has a unique solution. 7. Power-series expansions for ^-trajectories Since A-trajectories are parts of real algebraic curves, they have locally con- vergent power-series expansions about any (nonsingular) point of the curve. We derive two power-series expansions: the first is for an arbitrary ^-trajectory of a standard form problem in terms of a parameter measuring Euclidean distance in Legendre transform coordinates. The second power-series expansion is for the central ^-trajectory of a canonical form problem at the center, and uses the objective function value as the power-series parameter. Theorem 7.1. Given a strict standard form linear program minimize (c, x), Ax = b, x>0, and feasible point x0 = X0e with x0 > 0. Set c* = nA±(c), and parametrize the A-trajectory TA(x0 ; c) using Legendre transform coordinates by (7.1) 0H(x,) = 0H(xo)-/c\ Then the power-series expansion oo (7.2) x, = xo + £v/ k=\ has coefficient vectors v = vjt(c*) computed recursively as follows. Let Vk = diag(v^), and (7.3a) yx=X0n{AXo)±(X0c), (7.3b) License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE NONLINEARGEOMETRY OF LINEARPROGRAMMING. II 555 The recursion step determining vk and (7.4a) v, = -X0n{AXo)±lxoJ2n^X^V^) (7.4b) •*--^T'ÊV*-r 7=0 The matrices <¡>kare diagonal matrices. Proof. Define Xt = diag(x(), and the power-series expansion (7.2) to be deter- mined is equivalent to oo (7.5) *, = *o + EF/ Zc=l where the Vk are diagonal matrices. The vector x( - x0 lies in the subspace A = {x: Ax = 0} ; hence (7.2) yields (7.6) nA±(Vke)= 0 for k > 1. Define the diagonal matrices oo (7.7) ^r'^'E*/- Zc=0 Now (7.5) gives x;x = x-1 [i + x;x [j:vkt^ Kk=\ since diagonal matrices commute. Comparing the last two equations shows that Evaluating coefficients of powers of t in this formula yields Zc-l ^ + E^Av*-7,-1 = 0' ***■ 7=0 This yields the recursion (™) ^- since diagonal matrices commute, and proves (7.3b) and (7.4b). License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 556 D. A. BAYER AND J. C LAGARIAS Using the definition of Legendre transform coordinates and the definition (7.7) one obtains -v vk+Ev e k=l OO = 4>H(x0)-nA±\T(X-x®ke)tk\. Comparing this with (7.1) and equating powers of t gives (7.9a) nA±(X~X^xe) = c, (7.9b) nA±(X~X<í>ke)= 0, allzc>2. We have ®x = -X~x Vx by (7.8) so that (7.10) nA±(XQ2Vxe) = nA±(XQ 'o.e) = c*. The condition (7.6) asserts that v( = r^e is in A± , and Theorem 3.6 now shows that (7.10) has a unique solution which is given by Vl = X0n(AX0)± (X0C ) ' and this proves (7.3a). Next, for k > 2 substitute (7.8) into (7.9) to obtain (7.11) nA± (X~2 Vke) = - ¿ nAX(X^ Vk_,e). 7 = 1 Now yk = Vke is in A , so applying Theorem 3.6 to (7.11) yields *k = -X0*(AX0)± I X0 Ë nA-(X^jVk-je) j ' which is (7.9a) and completes the proof. (Note here that The formulae of Theorem 7.1 simplify considerably when the initial point x0 is e, as happens in a canonical form linear program. The recursive formulae for k = 1 are then (7.12a) Vj = -c*, (7.12b) 1=-^. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE NONLINEAR GEOMETRY OF LINEAR PROGRAMMING. II 557 and for k > 2 are (7.12c) (7.12d) In particular only the single projection nA± is involved in computing each yk . Now we derive a power-series formula for the central ^-trajectory of a canon- ical form linear program about the point e, using the value z = (x, e) of the objective function as the power-series parameter. The remarks after Theorem 6.2 imply that the objective function value parameter z is a monotone decreas- ing function of the Legendre transform coordinate parameter t. Theorem 7.2. Given a canonical form linear program in R" ' minimize (c*, x), Ax = 0, (e,x) = n, x>0, such that e = (1,1, ... , 1) is feasible and c* is in [¿f] . Define z = (c* ,x), and parametrize the central A-trajectory TA(e;c*) as {x(z)}, with x(0) = e. Then the power-series expansion around z = 0 is given by * k (7.13) x(z) = e + J2*lzkr~ k=\ where the coefficient vectors v* =v£(c*) are computed recursively as follows. Let Vk = diag(v^) and &0 = I, initialized with (7.14a) ax = -(c*,cTx, (7.14b) v* = -a,c*, (7.14c) &* = -V*. The recursion step determines ak, v*, and the diagonal matrices k-\ (7.15a) ak =a,(c*, ¿^C,e), X 7= 1 ' (7.i5b) v; = -^,Jx;*;Cy«]-a^' ,7=1 k-\ (7.150 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 558 D. A. BAYERAND J. C. LAGARIAS Proof. By definition x(z) is that point on the central ^-trajectory for which (7.16) z = (c,x(z)), and x(0) = e. Since x(z) is on the central ^-trajectory one has (7.17a) where (7.17b) /(z) = f>tz* Zc=0 converges in a neighborhood of z = 0 since oo / = f(z) = J>kz* . Zc=0 This gives a conversion formula from a power-series in t to one in z. Before beginning the calculations observe first that since x(z) is in [¿f] , (7.13) implies that (7.18) «r¿n(0-V Substitute the power-series expansion (7.13) into (7.16) and equate powers of z to obtain (7.19a) (c*,v;> = l, (7.19b) (c*,v*) = 0, allzc>2. To evaluate H(x(z)), let X* = diag(x(z)) and Vk = diag(v*). The power- series expansion (7.13) is equivalent to Zc=l Set oo (7.20) {K)~l=T,*lzk> k=0 and one easily finds as in Theorem 7.1 that O^ = I and (7.21) *î = -Ë*X-r all/c>l, 7=0 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE NONLINEARGEOMETRY OF LINEAR PROGRAMMING.II 559 which proves (7.14c) and (7.15c). Now compute r*\~l (f>H(x(z)) = n{^]±((Xz) e) k=\ J OO * \ k k = \ Equating this with (7.17) term by term yields q0 = 0 and (7.22) akc = n{^]± (fl>*e), all k > 1. Now (7.21) gives O* = -V* hence 0*e = -v* so that the last equation for zc= 1 yields (7.23) axc =-n[^]±(y*x). Multiplying this equation by (c*)T and using (7.19a) yields Q1(C*,C*) = -(C*,7T[^]X(VÍ)) = -(C*,V¡) = -1. This proves (7.14a). Also n,A,±(v*) = v*, so that (7.23) gives v* = -a,c*, which proves (7.14b). To derive the recursion step, multiply (7.22) on the left by (c*) to get (7.24) afc(c*,c*) = (c*,7r[^]±( x 7=0 ' using (7.21) and the fact that c* is in [¿f] . Now we note that using (7.18) and (7.19b). Substituting this in (7.24) yields k-\ X 7=1 ' which proves (7.15a). Finally (7.21) and (7.22b) give k-l -akc' = n{^(Vke) + £ n[&]±(&k_x Vje). 7 = 1 Using (7.18) this yields , >. <--n[^\È^i-Jvj'A-akc'' which proves (7.15b) and completes the proof, o License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 560 D. A. BAYER AND J. C. LAGARIAS The formula / = Y^kLoakz expresses t in terms of z. By reversion of power-series there is an inverse formula oo z = z(t)= J2ß/- k=0 In fact one has z = (c ,xt) = U* ,e + Y2yktk\ X Zc=l ' fc=l so that ß0 = 0 and ßk = (c*,yk), all/c>l, where the v^ may be computed using the formulae (7.12) with A replaced by One can consider "higher-order" analogues of Karmarkar's projective scaling algorithm, based on taking steps following truncated power-series expansions to the central ^[-trajectory. The choice of the power-series parameter plays a critical role in the performance of such algorithms. A second-order power-series implementation of these ideas, using the coordinate for which the objective function decreases (locally) most rapidly as the local power-series parameter, is described in [AKRV]. An average 25% decrease in the number of iterations of this second-order method compared to the projective scaling method was empirically observed on a standard set of 30 test problems. Other power-series implementations are described in [KLSW]. 8. A -TRAJECTORIES FOR FULL RANK LINEAR PROGRAMS It is possible to define ^-trajectories for an arbitrary linear program. We consider in detail the special case of a linear program in R" in inequality form: J minimize (c,x)-c0 ( ' \(*j,x)>bj, l License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE NONLINEAR GEOMETRY OF LINEAR PROGRAMMING. II 561 describe the image {J(x) : x e R" } , and which has an objective function (c, y) - c0 for which (8.3) (c,J(x))-c0 = (c,x)-c0 for all x e R". We define the A-trajectory TA(x0;c,H) to be the pullback under J-1 of the ,4-trajectory of the standard form problem through J(x0), which is J~ (T^J^îc^b)). To verify that this definition makes sense we must show that we can find a suitable J and (c,y) - c0, and that T^x^c,!!) is well defined independent of the choice of J and (c,y) -c0. A suitable mapping y = J(x) is given using the slack variables: (8-4) yj = (aj,x)-bj, since the transformed inequality constraints are now (8.5) yj >0, l and its Hessian is (8.8, V»AW.^W.g___lV,J which is positive definite for all x e Int(PH). At a given point x0 the Taylor- series approximation to /H(x) is (8.9) /H(x) = /H(x0) + (V/H(x0),x-x0) + i(x-x0)V/H(x-x0) + O(||x-x0||3). Now choose an affine rescaling of variables to make the quadratic term in the Taylor-series spherical. Set x = J(x) with J(x) = K(x- x0), where K is an invertible matrix chosen so that H = V2fH(x0) = KTK, License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 562 D. A. BAYER AND J. C. LAGARIAS which is always possible since H is a positive-definite symmetric matrix [HK, p. 247]. The transformed potential function has the Taylor-series ex- pansion: /H(t) = /H(x0) + ((K~X)VfH(x0),x) + (x,x) + 0(\\x\f). Then (8.10) The affine scaling direction field is the pullback under J ' of the negative of the transformed objective function direction -(K ) c, which is (8.11) yA(x0;c,rl) = (J-X)t(-(K-X)Tc) = -K'\k~x)tc = -(vVh(x0))_1c. The ,4-trajectory TA(x0;c,H) is obtained by integrating this vector field starting at x0. A third definition of ,4-trajectories uses Legendre transform coordinates: (8.12) 7^(x0;c,H) = {xe Int(PH): (8.13) 4>H 4>H Rel-Int(CH) ^— Int(Cp) Here J* is the adjoint mapping to J, defined by (J*(y),x>= License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE NONLINEAR GEOMETRY OF LINEAR PROGRAMMING. II 563 for all x e R", y € Rw. Theorem 3.4 also asserts that the mappings are one-to-one and onto in (8.13). By Corollary 4.1a the image of the ^-trajectories 7^(y;c,^,b) under ñis a family of parallel lines with slope nA±(c). Since J* is affine and one-to-one its image in R" is also a family of parallel lines. To show these lines coincide with those given by definition (iii) it suffices to prove that the slope of these lines is c. By definition of the adjoint we have (J*(^±(c)),x) = (^(c),J(x)-J(0)), for all x € R". Since J(x) - J(0) lies in A , (nA±(c),J(x)-J(0)) = (c,J(x)-J(0)). Comparing this with (8.3) yields (J*(nA±(c)),x) = (c,x)+cx, for all x e R" , which implies that cx = 0 and J*(nA± (c)) = c, completing the argument. (ii) o (iii). We compute the affine scaling vector field vA = v^3)(x;c,H) as- sociated to the third definition, which is defined by V/H(x + eyA) = 4>H(x) -ec + 0(e2). This just asserts that V v^-(v2/H(x)r'c. This coincides with (8.11), so that the second and third definitions are equiva- lent. D One can similarly define ^-trajectories for a general linear program of the form (8.1), which may have a lower-dimensional polytope of feasible solutions but is assumed to be of full rank. The analogue of the first definition above is as follows. The full rank condition guarantees that the slack variable transfor- mation J: R" -> Rm' given by y} = (»,•, x) - bj, l License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 564 D. A. BAYER AND J. C. LAGARIAS One can define ^-trajectories for nonfull rank linear programs, using (8.14). In this case the resulting ^-trajectories are not curves but are higher-dimensional objects; we omit the details. 9. Central ^-trajectories for full rank linear programs We define and study central ^-trajectories for an inequality form linear pro- gram: f minimize (c,x)-c0, 1 ' \(*j,l>bj, l For any x in Rel-Int(PH) one has License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE NONLINEARGEOMETRY OF LINEAR PROGRAMMING.II 565 This gives (9.3) Mx) = "g(e,x)-è;+g(c,x) + è-- Consequently if x e TA(c,H') then 0H,(x) = íc*, = ic* which with (9.3) shows that (9.5) TA(c, H) = {x,; Then x^ = limí_>ooxí exists. Set p = (c,xj, and let H' be H together with the constraints (9.6a) (c,x)>p, (9.6b) -(c,x)>-p,. Then x^ is the center of PH,. Furthermore PH, is the unique face of PH con- taining x^ in its relative interior. Proof. If c* = nD (c) = 0 then TA(c, H) = xH is the center of H , and xf = xH so x^ = xH . Also (c,x) is constant on H so the constraints (9.6) are singular constraints, hence PH, = PH and the theorem holds trivially in this case. Next suppose that c^, ^ 0, so that TA(c;H) is a curve by the full rank hypothesis. Since Legendre transform coordinates only become unbounded ap- proaching dPH = PH - Rel-Int(PH), and since x( parametrizes an algebraic curve, the limit x^ exists. Let Hn denote the set of nonsingular constraints of H , which we may suppose without loss of generality are (a,., x) > b,, l License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 566 D. A. BAYERAND J. C. LAGARIAS Then the feasible direction subspace Z)R = [am.+1, ... ,am]x . The added con- straints (9.7) are singular constraints of H' and its feasible direction subspace is DH' = [am* + l ' ••• 'am'C3 = [am- + l ' ••• >am'CH] ■ Hence nD i = 7c.c.,±nDH, since c* e DH implies that n,e.)± commutes with nD . Consequently by definition of x we have *DHA(X/)= V)^W^h(x,))) = V)^ic*) = ° Then (9.7) ^H,(xœ)=Hrn^H,(0H(xr)) = O. The definition of p guarantees that x^ is in PH,, and (9.7) implies that xœ must be in Rel-Int(PH,) and is the center xH, of H'. Finally the constraints (9.6) are just the equality constraint (c,x) = p. This constraint does not contain any relative interior point of PH ; hence the con- straints H' must cut out a face of PH , so PH, is a face of PH . Each point x of PH lies in the relative interior of a unique face of PH , so the theorem follows, o Since the range space of RH of the Legendre transform coordinate mapping is the relative interior of a cone, it follows that in (9.5) the limiting value t~ must be either 0 or -co. If t~ = 0 then üm;_>0+x does not exist, and the corresponding central trajectory is unbounded in the linear program's coordinates. 10. Central trajectories and linear programming duality There is a simple rational mapping from the central trajectory of a linear program to the central trajectory of its dual linear program. This mapping appears in Osborne [Os, Theorem 2.2], where it is stated in terms of logarithmic barrier functions. Consider a linear program in R" in inequality form: /minimize (c,x), {P) \ATx>b, where A = [a,, ... ,am]. The corresponding dual linear program in Rm is {minimize - (b, y), Ay = c, y>0. We say that the dual linear programs (P) and (D) are transverse if both objec- tive functions are nonconstant on their polytopes of feasible solutions. In the case that (P) has an interior feasible point the transversality condition is that c^O and n{AT)±(b) #0. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE NONLINEARGEOMETRY OF LINEAR PROGRAMMING.II 567 Let r|f) denote the positive half of the central trajectory of (P), defined by T{+P)= {x(t): where H^ denotes the constraints of (P), and 7j denotes the positive half of the central trajectory of (D), defined by Tf} = {y(t): 0Ho(y(O)= i^x(b), 0 < t < oo}, where Hß denotes the constraints of (D). Theorem 10.1. Let (P) and (D) be a pair of dual linear programs such that (P) has an interior feasible point and the pair (P) and (D) are transverse. If x(t) e r|P) and u(t) = ATx(t) - b are its corresponding slack variables then the vector y(t) e r| ' satisfies (10.1) Uj(t)yj(t) = l/t, l (10.2) *Hp(x(t))= -E (>.,,(/))-0, = -Mi)'1 = -tC where by definition -i _ /_1_ _1_ 1 \ UW -\uAt)'uM)'--'u(t))- Now define y = (l/r)u(í) v-l , and observe that y > 0 and Ay=-(l/t) ^Hfl(y) = -nAAY~le) = -tnA±(a(t)) = -tnA±(Ax(t) -b) = tnA±(b). Thus y = y(í) e t[D) . D As t —»oo the point x(f) e rj approaches an optimal solution x^ of (P), and y(t) e T+ ' approaches an optimal solution y^ of (D). To see this one need only observe that (10.2) implies that (y^ ,xoo) are both feasible and satisfy the complementary slackness conditions: (10.3) yjUj = yj((aj,x) -b;) = 0, l License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 568 D. A. BAYER AND J. C. LAGARIAS where c = (c,, ... ,cn) and c, ^ 0. Then using (10.1) one obtains such a rational mapping ô given by 1 (ax,x)-bx S{P\x) = 0H(X), 1 L(am,x)-¿z„ One can also find a rational mapping ¿( '(y) which when restricted to r|D) maps it to TfK One defines t by l¡t = o{D)oó{P)(x) x, xeT (P) The combined primal-dual linear program (PD) on Rm+" is minimize (c, x) - (b, y), ^4rx>b, (PD) Ay = c, y>0. Let HpD denote the set of constraints of (PD). It is easy to check that the Legendre transform coordinate mapping for (PD) is *"">\[y\) l^HD(y). ' and that the central ^-trajectory TpD for (PD) is PD {[yj^Qyj] '[^(b)]/ Hence the central ^-trajectory of (PD) projects onto the central ^-trajectories of (P) and (D), via orthogonal projections onto the x-variables and y-varia- bles, respectively. The algebraic correspondence given by Ö (x) associates the x coordinate of a point (x,y)r on the central ^-trajectory of (PD) to the y-coordinate. By Theorem 10.1 points on the central trajectory (x(p),y(p)) of the com- bined primal-dual linear program (PD) satisfy the system of equations A Tx- u = b, Ay = c, yjUj = p, l with 0 < p < oo, where all u > 0. The path (x(p),y(p)) is a special case of the parametric logarithmic barrier function trajectories studied by Megiddo [M]. It can be viewed as a homotopy path in the parameter p, and Kojima, License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE NONLINEAR GEOMETRYOF LINEAR PROGRAMMING. II 569 Mizumo and Yoshise [KMY] develop a polynomial-time linear programming algorithm that follows this path, starting with p = 1 and making p —>0. 11. vi-TRAJECTORIES AS ALGEBRAIC CURVES ^(-trajectories are pairs of real algebraic curves. In this section we give poly- nomial ideals of relations that cut out these curves. For any ideal / in the polynomial ring C[xx, ... ,xn] let V(I) = {(xx,... ,xn) eC": f(xx,... ,xn) = 0for all f el} denote the algebraic variety cut out by the ideal /. We can explicitly write down an ideal of relations satisfied by points on an ^-trajectory. Consider a linear program in R" in inequality form: ( minimize (c, x), ( ' \ License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 570 D. A. BAYER AND J. C. LAGARIAS Conversely if x is an interior point of P.p) for which there is a ( complex-valued) solution (x,y,u,p) of these equations, then x is on the central A-trajectory. Proof. Let V(A, b, c) denote the algebraic variety determined by the relations (11.2). Whenever x is a point on the positive part of the central trajectory of (P), then there is a real point (x,y,u,p) on V(A,b,c) by Theorem 10.1, having y > 0 and p > 0. The proof of Theorem 10.1 also shows that if x is a point on the negative part of the central trajectory of (P), then there is a real point (x,y,u,p) on V(A,b,c) having y<0 and p<0. Conversely if (x,y,u,p) is a point on V(A,b,c) and x€lnt(P(/))) then x must be on the central trajectory. To see this, one calculates that the equations (11.2) imply that License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE NONLINEAR GEOMETRY OF LINEAR PROGRAMMING. II 571 consider the hyperplane at infinity z = 0 in p"+2m+1(ç) intersected with the homogeneous ideal A x - u = bz, Ay = cz, yjUj = pz, I It is easy to see that for full rank A that V(A,b,c) contains an (m - n)- dimensional projective space determined by z = 0, y = 0, A x - u = 0 ; it contains many other components of dimension > 2 as well. It remains an open problem to give generators for an ideal cutting out exactly the irreducible curve V(TA). 12. LAGRANGIAN DYNAMICAL SYSTEMS PRODUCING ^-TRAJECTORIES The Legendre transform plays an important role in classical mechanics; it converts a dynamical system given in Lagrangian form to one in Hamiltonian form, see [Ar, Ln]. This conversion replaces a system of n second-order dif- ferential equations in n variables (Lagrange's equations) with a system of 2« first-order differential equations in 2n variables (Hamilton's equations). The simple form of ^-trajectories and P-trajectories in Legendre transform coordi- nate space suggests that they might be described by a simple dynamical system in Lagrangian or Hamiltonian form. This is the case for ^-trajectories. Consider linear programs in R" in the inequality form: Í minimize (c,x), [ } \(aj,x)>bj, \<} License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 572 D. A. BAYER AND J. C. LAGARIAS together with (12.2b) è>= Tti(ii)' l^^n- The first Lagrangian dynamical system we consider has position variables q in R" and the Lagrangian m (12.3) L1(q,q) = (c,q)-^log((a.,q)-Z7.). 7=1 We show that the q-trajectories of this Lagrangian system are the complete set of ^-trajectories for the linear program (12.1), having a fixed objective function (c,x>. Theorem 12.1. Suppose that the set H of linear program constraints (12.4) (a7,x)>¿>7, l (12.6) p = -E7¿^7r^H(q), where (12.7) ^( License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE NONLINEAR GEOMETRY OF LINEAR PROGRAMMING. II 573 But this is exactly the definition of an ^-trajectory for the objective function (c,x) for this linear program, by (8.12). Since c0 is arbitrary we get all A- trajectories for the fixed objective function (c,x) this way. D We convert this Lagrangian dynamical system to an equivalent Hamiltonian dynamical system. A Hamiltonian dynamical system is specified by a Hamil- tonian function H(p, q) defined on (a subset of) R " and has variables (p, q) with p = (px, ... ,pn)T ■ Rn is usually called phase space. This system evolves according to Hamilton's equations, which are n2OA (p, = -dH/dq¡, l Wx,¥l) trUp,^,. dq,dp, A Hamiltonian dynamical system on R " is said to be (globally) completely integrable if there is an open subset S of phase space R which is an invariant subset of H (i.e., it is a union of trajectories satisfying (12.9)) and there exist n functions FX,F2, ... ,Fn defined on S such that (a) {H,F,} = 0 for 1 < i < n , (b) {F,,Fj} = 0 for 1 (12.10) H(p,q) = Yàp,q,-L(q,q). 7=1 In this formula one needs to express the q, in terms of the (p, q) variables. To do this one takes the defining relations (12.11) Pi = dL/dq,, 1 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 574 D. A. BAYER AND J. C. LAGARIAS The Hamiltonian is then expressed explicitly as n (12.13) H(p,q) = J£pif(p,q) - L(q,f(p,q)). 1=1 This conversion method works in principle but usually not in practice since the inversion (12.11) cannot usually be specified in closed form. (In addition there may be singular points where the inversion cannot be done.) In the case of the Lagrangian L,(q, q) in Theorem 12.1 we can carry out this inversion and prove the following result. Theorem 12.2. Suppose that the constraints H given by (aj ,x)>bj, 1 < j < m, have a bounded polytope of feasible solutions with nonempty interior. Then the Hamiltonian Hx (p, q) corresponding to the Lagrangian m ¿i fa »q) = Proof. The hypotheses of Theorem 12.1 apply. There we observed that p = dL/dq = 0H(q), so that q = (p^x(p). Substituting the formulae q, = (p^ (p), into (12.13) yields formula (12.14). It remains to prove complete integrability. Take {c , ... ,c } to be a basis of c1, and set Fk(p,q) = (c{k),p), l Fn(p,q) =Hx(p,q). We claim these functions satisfy conditions (a)-(c) for complete integrability. We need only verify (b) and (c) since {Hx ,F,} = {Fn,F,} . It is clear that {Fki,Fk2} = 0, l 1 " k) ^f\dp, dp, dq,dq," dq, dp,dpj /=1 E(k) 7k) iAk)i (k) , A c.c\cic = (c , c) = 0 i=i License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE NONLINEARGEOMETRY OF LINEAR PROGRAMMING. II 575 for 1 < k < n - 1, which proves (b). To verify (c) we show that {dFk: 1 < k (12.15) ^«-1 dF.n-\ dFn where the entries in * are functions of (p, q). The linear independence of the rows of (12.15) at all points of R " is clear. This proves (c). □ The last two theorems imply that the q-trajectories of the Hamiltonian Hx(p,q) are exactly the ,4-trajectories of the linear program (12.5). This can also be verified directly from Hamilton's equations for Hx (p, q). Hamilton's equations are (12.16a) P = c, . dH, q dp -i, -i, = (12.17) L2(q,q,q,q) = ^(q + 4>q+4) - Elog((av'^ ~ ^' 7=1 The q-trajectories of this dynamical system give all of the ,4-trajectories. Theorem 12.3. Suppose that the set H of linear program constraints (&j,x)>bj, 1 < j < m, License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 576 D. A. BAYERAND J. C. LAGARIAS has a bounded polytope of feasible solutions in R" with nonempty interior. Then each trajectory {q(t) : - oo < t < 00} of the Lagrangian dynamical system with Lagrangian 1 • " l2 = 2 fa+q>q+q) - El08(fa> »q>- bj) 7 = 1 is an A-trajectory of a linear program with constraints H and some objective function (c, x), and conversely all A-trajectories for such linear programs with all possible objective functions (c, x) arise as q-trajectories of this Lagrangian dynamical system. Remark. In the case c = 0 these trajectories are constants. We make the theo- rem hold in this case by defining ^-trajectories to be points when c = 0. Proof. This is an easy computation. We see that <9L, , . 9L, ^=q + q, ^=0, dL2 dL2 . ^ = 0H(q)> j(=*+*- Hence Lagrange's equations of motion are ^(0H(q)) = q+q> ^fa+^ = 0- Consequently one has q + q = c where the c are arbitrary constants of integra- tion, and License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE NONLINEAR GEOMETRY OF LINEAR PROGRAMMING. II 577 Proof. One has p = é>L/cZq = Fk =pk, 1 < k < n, Fn+k=PxPk+x-pk+xPx, l dF2n-l p2 p, • •• 0 P2-Px- dFIn Pn 0-p, Pn 0- Px-Qx-Pn-Qn License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 578 D. A. BAYER AND J. C. LAGARIAS where I is an n x n identity matrix and O denotes a zero matrix (of the appropriate size). It is clear that this matrix has full row rank whenever px ^ 0. Finally we note that the open set R4" - {(p,p,q,q): p, = 0} is an invariant set for H2. Indeed, since Fx = p, is an integral of the motion, each level set p, = constant is separately an invariant set. D What happens when the linear program does not have a bounded, full-dimen- sional polytope of feasible solutions? In the case that the feasible solution poly- tope PH is an unbounded full-dimensional polytope, the Lagrangians L, and L2 are defined identically as in the bounded case, and the trajectories corre- sponding to ^-trajectories occupy only part of the phase space. The image of PH under the Legendre transform coordinates (pH(PH) occupies an open full- dimensional pointed cone in the p-coordinates (by Theorem 3.5) and trajecto- ries having p-coordinate inside this cone correspond to ^-trajectories. These trajectories are not defined for all time but only for a time interval (t0,tx), depending on the objective function, with some p-variable diverging at finite time. These Hamiltonian systems are still completely integrable in an appro- priate open invariant subset of the phase space. The remaining case where PH is a lower-dimensional polytope can in principle be reduced to one of the full- dimensional polytope cases by eliminating variables, i.e., restricting the problem to the lower-dimensional flat in R" spanned by PH . There are several open questions suggested by these dynamical systems. First, are there analogous completely integrable Hamiltonian dynamical systems de- scribing P-trajectories? Second, these systems identify ,4-trajectories with q- trajectories, which correspond to velocities in the physical interpretation. What connection (if any) to the linear programming problem do the position trajecto- ries ( q-trajectories) have? Third, for a linear program with a bounded polytope PH in which both (c,x) and -(c,x) have unique optimal solutions, all the q-trajectories of the Lagrangian L,(q,q) have unique limiting velocities q(oo) as i —»+00 and q(-oo) as t —►-oo (which correspond to optimal solutions of these two linear programs). Hence they exhibit forward and backward scat- tering. Is there a scattering theory interpretation of what this dynamical system is doing in the q-variable space? Appendix A. Abstract Legendre transform coordinates The Legendre transform coordinate mapping (aj ,x)>bj, 1 < j < m, having the polytope of feasible solutions PH and feasible direction subspace DH . Let H;| denote the set of nonsingular constraints, i.e., these constraints that do not hold with equality in PH , which by renumbering constraints if License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE NONLINEAR GEOMETRY OF LINEAR PROGRAMMING. II 579 necessary may be taken to be fHn(x + ey) = fHn(x) + e((dfHn)x,v) + 0(e2), as e —»0. Let D^ be the subspace polar to DH in (R")*, defined by DPH= {yeR")*: (y,x) = 0 for all x e DH}, and let ñ^ : (97)* -> (R")*/D^ be the quotient mapping. The abstract Legen- dre transform coordinate mapping 4>H(x)=ñDPH((dfHn)x)- The coordinate-free analogues of the results of §3 are as follows. Here we regard the constraint coefficients a as elements of (R")*. Theorem A. 1. The abstract Legendre transform coordinate mapping 4>H(x) has domain Rel-Int(PH) andränge Rel-Int(CH) where CH =7fDP(CH ) and CH = R+[-a,, ... , - am.] in (R")*. // H is of full rank then dim(CH) = dim(PH) and 0H is a one-to-one mapping. Let J(x) = Lx + m be an affine mapping for R" to R . Define its adjoint mapping J* to be the linear mapping (R )* —►(R")* given by «Ah KDpoJ Rel-Int(CH) <— - Rel-Int(CJ(H)) License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 580 D. A. 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